Let's make two initial comments to muddy the waters! 1) Accepting some version of a correspondence theory of truth -- e.g. accepting that a true proposition is made true by the existence of a corresponding fact -- doesn't ipso fact make you a realist in your ontology. It will obviously depend what you think about facts ! (You could still be an idealist like Berkeley, and suppose the only facts are ultimately those involving God, other spirits, and their ideas.) 2) Accepting that our knowledge of the world depends on a lot of processing of data by the brain using built-in cognitive mechanisms doesn't make you an anti-realist in epistemology. You could still hold that when those processes are working reliably, they successfully give you epistemic access to facts that obtain independently of you and your cognitive mechanisms. I'd say that talk of a "correspondence theory of truth", "realism about ontology", "conceptual frameworks", and "epistemological anti-realism" is all far too slippery...

You need to be careful to distinguish things and ideas here. Is the question about square circles or about the idea of a square circle ? Compare: there are no such things as unicorns . It would plainly be wrong to say that they "neither exist nor don't exist": unicorns definitely don't exist! But the idea of a unicorn exists and seems coherent enough. Indeed, we are tempted to suppose that there could have been things that fitted the idea. Likewise, there are no such things as round squares . Like unicorns, round squares definitely do not exist. But this time, though we can frame the idea of a round square -- we grasp that something counts as a round square if it is round and a square! -- the idea is a self-contradictory one in the sense that nothing can possibly count as fitting our idea here.

Consider an example from Frege: the direction of the line L is identical to the direction of the line M if and only if L is parallel to M. That's true. But how should we read it? Do we read it as explaining the notion of being parallel in terms of the identity of two abstract objects, i.e. two directions? Or do we take it the other way about, as partially explaining talk about two abstract objects, directions, in terms of the already-understood notion of lines being parallel? There's lots to be said for taking it the second way, as introducing reference to certain abstract objects in terms of something more familiar. Likewise: the meaning of "gorse" is identical to the meaning of "furze" if and only if "gorse" and "furze" are synonymous. That looks true too. But how should we read it? Do we read it as explaining the notion to synonymy in terms of the identity of two abstract objects, meanings? Or do we take it the other way about, as (hopefully) partially explaining talk about two...